class: center, middle, inverse, title-slide .title[ # Class 2a: Review of concepts in Probability and Statistics ] .author[ ### Business Forecasting ] --- <style type="text/css"> .remark-slide-content { font-size: 20px; } </style> ### Summary - In the last class: - We discussed the organization of the course - We overviewed forecasting methods - We learned about methods of qualitative forecasting - *Reference:* Forecasting Methods and Applications, chapter 1 - This set of classes: - We will start learning about exploratory analysis preparing the forecast - We will learn about various **data types** - We will learn how to **summarize data graphically** - We will learn how to **summarize data with summary statistics** - We will learn about **comparisons and associations** - *Reference:* Forecasting Methods and Applications, chapter 2.1-2.4 --- ### Scenario - Nowadays, many online pharmacies appeared which write prescriptions and make drugs subscriptions - Example in Mexico: *Choiz* -- - At the same time, a new wave of very effective anti-diabetes drugs appeared which help to lose weight - Example: *Ozempik* -- - You are consulting a business which wants to provide subscription services for these drugs in Mexico -- - Your boss asks you to do exploratory market research for potential sales forecast --- ### Parameters vs Statistics - You need to know how many people in Mexico have diabetes -- #### Parameter - Call `\(\mu_d\)` the proportion of Mexican population which has diabetes - Usually the parameter is an **unknown** number **describing the whole population** - You want to learn what it is - In our example, `\(\mu_d\)` is a parameter that you want to learn - More generally, parameter describes an aspect of the entire population -- #### Statistic - But you don't have data on the whole population. At best you can get a sample from a survey - So you will try to estimate this parameter with sample - Statistic is a **guess of the parameter** which can be **calculated from the sample** - You will calculate a statistic `\(\hat{\mu}_p\)` which is the proportion of diabetics in the sample --- ### Parameters vs Statistics - What is population, sample, parameter and statistic in the following examples? -- - You want to know the probability that a user who got a match on tinder will go out on a date with that person. You survey 1000 users and ask them about each match they got if they went on a date. You then calculate the share of dates which ended up in a match for these users. -- - You want to know what whether starbucks baristas are faster than Cielito Querido baristas. You go to 10 starbucks and 10 Cielito Querido and measure the time it takes to make a coffee. You then calculate the average time it takes to make a coffee in each of these chains. -- - You want to know the average age of people who go to the gym. You go to a gym and ask 100 people about their age. You then calculate the average age of these people. -- - You want to know the variance of internet speed during in Mexico City. You visit 500 households and calculate the variance of their internet speed. --- --- layout: false class: inverse, middle # Types of Data --- ### Longitudinal Data - Observations are collected for the same subject (entity) over a period of time - Same as time series data - Example: Tracking a company's annual revenue and number of employees over several years #### Longitudinal Data Example <div class="datatables html-widget html-fill-item" id="htmlwidget-f67acfb0ab41a4124c66" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-f67acfb0ab41a4124c66">{"x":{"filter":"none","vertical":false,"fillContainer":false,"data":[[2018,2019,2020,2021,2022],[50000,52000,55000,58000,60000],[50,55,60,65,70]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th>Year<\/th>\n <th>Revenue<\/th>\n <th>Employees<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":5,"searching":false,"initComplete":"function(settings, json) {\n$(this.api().table().container()).css({'font-size': '12px'});\n}","columnDefs":[{"className":"dt-right","targets":[0,1,2]},{"name":"Year","targets":0},{"name":"Revenue","targets":1},{"name":"Employees","targets":2}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[5,10,25,50,100]},"selection":{"mode":"multiple","selected":null,"target":"row","selectable":null}},"evals":["options.initComplete"],"jsHooks":[]}</script> --- ### Cross-Sectional Data - Observations are collected at a single point in time - Example: A survey of customers' satisfaction with a product and likelihood of repurchase at a certain point in time #### Cross-Sectional Data Example <div class="datatables html-widget html-fill-item" id="htmlwidget-5d3784c68f6bead26537" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-5d3784c68f6bead26537">{"x":{"filter":"none","vertical":false,"fillContainer":false,"data":[[1,2,3,4],[7,8,5,9],["Likely","Unlikely","Likely","Likely"]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th>Customer_ID<\/th>\n <th>Satisfaction_Score<\/th>\n <th>Repurchase_Likelihood<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":5,"searching":false,"initComplete":"function(settings, json) {\n$(this.api().table().container()).css({'font-size': '12px'});\n}","columnDefs":[{"className":"dt-right","targets":[0,1]},{"name":"Customer_ID","targets":0},{"name":"Satisfaction_Score","targets":1},{"name":"Repurchase_Likelihood","targets":2}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[5,10,25,50,100]},"selection":{"mode":"multiple","selected":null,"target":"row","selectable":null}},"evals":["options.initComplete"],"jsHooks":[]}</script> --- ### Panel Data - Combines both longitudinal and cross-sectional data - Observations are collected for multiple subjects over multiple points in time - Example: Tracking the annual revenue and number of employees of several companies over a few years #### Panel Data Example <div class="datatables html-widget html-fill-item" id="htmlwidget-19ebac2509a70925da32" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-19ebac2509a70925da32">{"x":{"filter":"none","vertical":false,"fillContainer":false,"data":[[2018,2018,2018,2019,2019,2019,2020,2020,2020,2021,2021,2021,2022,2022,2022],["A","B","C","A","B","C","A","B","C","A","B","C","A","B","C"],[50000,52000,55000,58000,60000,60000,63000,65000,68000,70000,70000,72000,75000,78000,80000],[50,55,60,65,70,100,105,110,115,120,150,155,160,165,170]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th>Year<\/th>\n <th>Company<\/th>\n <th>Revenue<\/th>\n <th>Employees<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":5,"searching":false,"initComplete":"function(settings, json) {\n$(this.api().table().container()).css({'font-size': '12px'});\n}","columnDefs":[{"className":"dt-right","targets":[0,2,3]},{"name":"Year","targets":0},{"name":"Company","targets":1},{"name":"Revenue","targets":2},{"name":"Employees","targets":3}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[5,10,25,50,100]},"selection":{"mode":"multiple","selected":null,"target":"row","selectable":null}},"evals":["options.initComplete"],"jsHooks":[]}</script> --- ## Q1 <div class="datatables html-widget html-fill-item" id="htmlwidget-9cddd6f0355441d6bf92" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-9cddd6f0355441d6bf92">{"x":{"filter":"none","vertical":false,"fillContainer":false,"data":[["Jan","Jan","Jan","Feb","Feb","Feb","Mar","Mar","Mar","Apr","Apr","Apr","May","May","May","Jun","Jun","Jun","Jul","Jul","Jul","Aug","Aug","Aug","Sep","Sep","Sep","Oct","Oct","Oct","Nov","Nov","Nov","Dec","Dec","Dec"],["Bitcoin","Ethereum","Dogecoin","Bitcoin","Ethereum","Dogecoin","Bitcoin","Ethereum","Dogecoin","Bitcoin","Ethereum","Dogecoin","Bitcoin","Ethereum","Dogecoin","Bitcoin","Ethereum","Dogecoin","Bitcoin","Ethereum","Dogecoin","Bitcoin","Ethereum","Dogecoin","Bitcoin","Ethereum","Dogecoin","Bitcoin","Ethereum","Dogecoin","Bitcoin","Ethereum","Dogecoin","Bitcoin","Ethereum","Dogecoin"],[60000,40000,10000,62000,41000,10500,63000,42000,11000,64000,43000,11500,65000,44000,12000,66000,45000,12500,67000,46000,13000,68000,47000,13500,69000,48000,14000,70000,49000,14500,71000,50000,15000,72000,51000,15500]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th>Month<\/th>\n <th>Cryptocurrency<\/th>\n <th>Market_Cap<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":5,"searching":false,"initComplete":"function(settings, json) {\n$(this.api().table().container()).css({'font-size': '12px'});\n}","columnDefs":[{"className":"dt-right","targets":2},{"name":"Month","targets":0},{"name":"Cryptocurrency","targets":1},{"name":"Market_Cap","targets":2}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[5,10,25,50,100]},"selection":{"mode":"multiple","selected":null,"target":"row","selectable":null}},"evals":["options.initComplete"],"jsHooks":[]}</script> -- **Panel data** - Multiple time observation per subject (currency) and multiple subjects --- ## Q2 <div class="datatables html-widget html-fill-item" id="htmlwidget-92734c20253fa4f1c619" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-92734c20253fa4f1c619">{"x":{"filter":"none","vertical":false,"fillContainer":false,"data":[["United States","China","India","Brazil","Russia"],[331,1439,1380,213,145],[21433,15308,3160,1848,1690],[246,904,560,126,116]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th>Country<\/th>\n <th>Population_Millions<\/th>\n <th>GDP_Billions<\/th>\n <th>Internet_Users_Millions<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":5,"searching":false,"initComplete":"function(settings, json) {\n$(this.api().table().container()).css({'font-size': '12px'});\n}","columnDefs":[{"className":"dt-right","targets":[1,2,3]},{"name":"Country","targets":0},{"name":"Population_Millions","targets":1},{"name":"GDP_Billions","targets":2},{"name":"Internet_Users_Millions","targets":3}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[5,10,25,50,100]},"selection":{"mode":"multiple","selected":null,"target":"row","selectable":null}},"evals":["options.initComplete"],"jsHooks":[]}</script> -- **Cross-sectional data** - Single (time) observation per subject (user), many subjects --- ## Q3 <div class="datatables html-widget html-fill-item" id="htmlwidget-d0aee4241f7a2c27f260" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-d0aee4241f7a2c27f260">{"x":{"filter":"none","vertical":false,"fillContainer":false,"data":[[2020,2021,2022,2023,2024],[20000,30000,40000,50000,60000]],"container":"<table class=\"display\">\n <thead>\n <tr>\n <th>Year<\/th>\n <th>Electric_Car_Sales<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":5,"searching":false,"initComplete":"function(settings, json) {\n$(this.api().table().container()).css({'font-size': '12px'});\n}","columnDefs":[{"className":"dt-right","targets":[0,1]},{"name":"Year","targets":0},{"name":"Electric_Car_Sales","targets":1}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[5,10,25,50,100]},"selection":{"mode":"multiple","selected":null,"target":"row","selectable":null}},"evals":["options.initComplete"],"jsHooks":[]}</script> -- **Longitudinal data** - Multiple (time) observations of a single subject --- ### Primary vs Secondary Data -- - **Primary data** is original data collected .red[directly from the source] for a .red[specific research purpose]. - Experimental data (if used by team designing the experiment) - Survey data (if used by survey team designing the surey) - It is customized for a particular research objective -- - **Secondary data** is data that has .red[already been collected by someone else] for a different purpose but .red[can be used for a new research question or analysis] - National statisics - death certificates - National surveys reused by researchers - Data from medical records - Data on stock market --- ### What kind of data is it? <small> - Surveys: A company conducts a customer satisfaction survey to gather feedback from its customers regarding their products and services. - Sales Reports: A business can analyze past sales data from previous years to identify trends and make strategic decisions. - Interviews: A researcher interviews individuals to understand their opinions on a particular topic, such as political preferences or healthcare choices. - Census Data: Government census data can be used by researchers to study demographic trends or population characteristics in a specific region - Observations: An ecologist observes the behavior of a particular species in its natural habitat to gather data for a research project. - Social Media Data: Companies can analyze social media posts and user engagement data from platforms like Twitter or Facebook to gain insights into customer preferences and sentiment. - Experiments: Scientists conduct laboratory experiments to test a specific hypothesis and collect data directly from the experiments. - Academic Journals: Researchers can review published studies and articles to gather data and insights related to their research topic. </small> --- layout: false class: inverse, middle ## Variable Types --- ## Variable Types We have two general types: .blue[Categorical] and .blue[Numerical] variables ### Categorical Variables - Variables that can be divided into one or more groups or categories. - **Ordinal:** These variables can be logically ordered or ranked. - *Variable:* Customer Satisfaction Survey Results - *Example:* Very Unsatisfied, Unsatisfied, Neutral, Satisfied, Very Satisfied - **Nominal:** These variables cannot be ordered or ranked. - *Variable:* Social Media Platforms Used - *Example:* Facebook, Instagram, Twitter, LinkedIn, TikTok, Snapchat --- ### Numerical Variables - Variables that hold numeric value and ordering is possible - **Discrete:** These variables can only take certain values - *Example*: Number of App Downloads from App Store - *Example*: Number of children you have - *Example*: Size of coke products: 0.33L, 0.5L, 1L, 2.25L <center> <img src=coke_sizes.jpg width="500"> </center> --- ### Numerical Variables - Variables that hold numeric value and ordering is possible - **Discrete:** These variables can only take certain values - *Example*: Number of App Downloads from App Store - *Example*: Number of children you have - *Example*: Size of coke products - **Continuous:** These variables can take any value within a range - *Example*: Time spent on a Webpage - *Example*: Exchange rate between MXN and USD -- - What's the main difference between ordinal and discrete? - We could say 1=Very unsatisfied, 2=Unsatisfied - But we cannot say that very unsatisfied has half of satisfaction of person who is just unsatisfied! - We can order, but these numbers don't have meaning in terms of distance between them --- ### Mexican Health Survey Representative sample of the Mexican population .red[n=37858] <div class="datatables html-widget html-fill-item" id="htmlwidget-23b73143271bb7543536" style="width:100%;height:auto;"></div> <script type="application/json" 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class=\"display\">\n <thead>\n <tr>\n <th>weight<\/th>\n <th>n_i<\/th>\n <th>p_i<\/th>\n <\/tr>\n <\/thead>\n<\/table>","options":{"pageLength":6,"searching":false,"initComplete":"function(settings, json) {\n$(this.api().table().container()).css({'font-size': '12px'});\n}","columnDefs":[{"className":"dt-right","targets":[1,2]},{"name":"weight","targets":0},{"name":"n_i","targets":1},{"name":"p_i","targets":2}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[6,10,25,50,100]},"selection":{"mode":"multiple","selected":null,"target":"row","selectable":null}},"evals":["options.initComplete"],"jsHooks":[]}</script> ] - Most values never repeat, so they have very low relative frequency --- ## Histograms **Solution**: Group similar values together - Construct intervals and show how many observations are in a given interval -- **Process** 1. Decide how many intervals 2. And how wide they are 3. Then calculate the absolute and relative frequencies of each interval 4. Plot it with bars -- --- **My approach** - I want `\(k\)` (example `\(k\)`=5) equal intervals -- - Divide the range of the data into `\(k\)` equal intervals -- - *Range* is max-min of the data -- ``` r # Calculate max and min max_value <- max(Health_data$weight) min_value <- min(Health_data$weight) # Calculate the difference range <- max_value - min_value ``` -- ``` ## [1] "Range= 190.8078 - 30.3745 = 160.4333" ``` -- - With 5 intervals, each will be 32kg wide -- - The first one starts at the minimum value (30.3745) -- - The last one ends at the maximum value (190.8078) -- - Calculate how many observations I have in each interval and what's the relative frequency --- ## Histograms - Midpoint represents middle of the interval - center of the bar - `\(P_i\)` is cumulative frequency: share of observations in this or smaller interval - *Example*: `\(P_{(62.46-94.55)}=0.911\)` - *Interpretation*: 91.1% of people have weight lower than 94.55kg .pull-left[ <!-- --> ] .pull-right[ <div class="datatables html-widget html-fill-item" id="htmlwidget-bcb7704b8d49e60f6149" style="width:100%;height:auto;"></div> <script type="application/json" data-for="htmlwidget-bcb7704b8d49e60f6149">{"x":{"filter":"none","vertical":false,"fillContainer":false,"data":[["30.37 - 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Helps to see the distribution and outliers - Is more always better? - With smaller intervals, histogram tends to the **probability density function** --- ## Probability Density Function (PDF) ### Definition - **Probability Density Function (pdf)** describes the probability distribution of a continuous random variable. - It **does not** give probability at a given value (this is always 0 for continous variable) - It shows which in which intervals that variable the most often appears - It is used to calculate the probability of the random variable being in a given interval - Area under it always adds up to 1 -- ### Example We have a random variable X representing the weight of adults in Mexican population. 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100, 100, 0.3)","marker":{"color":"rgba(255,127,14,1)","line":{"color":"rgba(255,127,14,1)"}},"error_y":{"color":"rgba(255,127,14,1)"},"error_x":{"color":"rgba(255,127,14,1)"},"xaxis":"x","yaxis":"y","frame":null}],"highlight":{"on":"plotly_click","persistent":false,"dynamic":false,"selectize":false,"opacityDim":0.20000000000000001,"selected":{"opacity":1},"debounce":0},"shinyEvents":["plotly_hover","plotly_click","plotly_selected","plotly_relayout","plotly_brushed","plotly_brushing","plotly_clickannotation","plotly_doubleclick","plotly_deselect","plotly_afterplot","plotly_sunburstclick"],"base_url":"https://plot.ly"},"evals":[],"jsHooks":[]}</script> --- ## Distribution Shapes: Modality <img src="data:image/png;base64,#C_2_slides_a_files/figure-html/unnamed-chunk-37-1.png" width="100%" /> --- ## Which is uniformaly distributed 1. weights of adult females 2. salaries of a random sample of people from CDMX 3. House prices in CDMX 4. birthdays of classmates (day of the month) --- ## Distribution Shapes: Skewness <img src="data:image/png;base64,#C_2_slides_a_files/figure-html/unnamed-chunk-38-1.png" width="100%" /> --- ### Age at death <center> <img src=Age_at_death.jpeg width="800"> </center> --- ## We want to know how many people weight more than 100kg --- ## Cumulative Distribution Function (CDF) The .blue[Cumulative Distribution Function] (CDF) gives the probability that a random variable X will take on a value less than or equal to a specific value. For a continuous random variable X with PDF f(x), the CDF F(x) is defined as: `\(F(x) = \int_{-\infty}^{x} f(t) \, dt = P(X \leq x)\)` Characteristics: - The CDF starts (for minus infinity) at 0 (minimum) - It approaches 1 as x approaches infinity (maximum) - It is non decreasing - It is right continuous --- ## Example 1: Standard Normal `\(F(-2) = \int_{-\infty}^{-2} f(t) \, dt = P(X \leq -2)=0.02\)` <!-- --> --- ## Example 2: Standard Normal `\(F(0.2) = \int_{-\infty}^{0.2} f(t) \, dt = P(X \leq 0.2)=0.58\)` <!-- --> --- ## Example 3: Standard Normal `\(F(3.2) = \int_{-\infty}^{3.2} f(t) \, dt = P(X \leq 3.2)=0.99\)` <!-- --> --- ### Empirical CDF We can do similar thing with our weight data. Intuition on how it comes up: <!-- --> <!-- --> --- ### Empirical CDF We can do similar thing with our weight data. Intuition on how it comes up: <!-- --> <!-- --> --- ### Empirical CDF `\(ECDF(x)=\frac{\sum I(w_i\leq x)}{N}=\frac{\text{Number of people with weight lower than x}}{N}\)` <small> - `\(I(w_i<x)=1\)` if weight of person `\(i\)` is lower than `\(x\)` (*Indicator Function*) - `\(N\)` is total number of people (*Sample Size*) - Share of people with weight lower than x </small> <div id="htmlwidget-a804cd50813264ff574a" style="width:800px;height:300px;" class="plotly html-widget"></div> <script type="application/json" 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Sans","size":26.567040265670411}},"hovermode":"closest","width":800,"height":300,"barmode":"relative"},"config":{"doubleClick":"reset","modeBarButtonsToAdd":["hoverclosest","hovercompare"],"showSendToCloud":false},"source":"A","attrs":{"345c3982660e":{"x":{},"type":"scatter"}},"cur_data":"345c3982660e","visdat":{"345c3982660e":["function (y) ","x"]},"highlight":{"on":"plotly_click","persistent":false,"dynamic":false,"selectize":false,"opacityDim":0.20000000000000001,"selected":{"opacity":1},"debounce":0},"shinyEvents":["plotly_hover","plotly_click","plotly_selected","plotly_relayout","plotly_brushed","plotly_brushing","plotly_clickannotation","plotly_doubleclick","plotly_deselect","plotly_afterplot","plotly_sunburstclick"],"base_url":"https://plot.ly"},"evals":[],"jsHooks":[]}</script> -- - So how do we calculate share of people with weight>100kg? `\(P(weight>100)=1-P(weight\leq100)=1-ECDF(100)\)` --- ### Exercises: - Review Exercises: - PDF 1: 1,2,3,4,5,6,7 - PDF 2: 14,15,16,17